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34. INTRODUCTION TO ABSOLUTE VALUE
Karl and Julia
live the same
distance from you
Suppose you live in the big white house shown below. Julia lives in the green
house to the right, and Karl lives in the blue house to the left.
Karl and Julia live in different places, but they both live the same distance
from your home. To visit Julia, you would walk to the right. To visit Karl, you
would walk to the left.
Suppose you were asked the question: “Who lives one mile from you on this
street?” You would respond by saying: “Well—two people. Julia lives one mile
this way, and Karl lives one mile that way.”
This section explores the idea of distance from home, under the guise of a
mathematical concept called absolute value.
blue
.
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KARL
distance from zero:
3 and −3
have the same
distance from 0
green
white
.....
←− 1 mile −→
..... ...........
......
...
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←− 1 mile −→
YOU
...
...... ...........
......
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JULIA
Now, think of “home” as being position 0 on a number line. Instead of talking
about distance from home, we’ll talk about distance from zero.
When you talk about distance from zero, it’s critical to remember that you can
walk away from 0 in two directions: to the right, or to the left.
Suppose you’re asked the question: “What numbers are three units from 0 on
a number line?”
The number 3 is three units from 0 ; it is 3 units to the right of 0 .
Also, the number −3 is three units from 0 ; it is 3 units to the left of 0 .
Opposites (like 3 and −3) always have the same distance from 0 : one is to the
right of zero, and one is to the left of zero.
←−−− distance = 3 −−−→←−−− distance = 3 −−−→
−3
EXERCISES
1.
3
On a number line, shade the specified number(s):
(a) all
(b) all
(c) all
(d) all
(e) all
(f) all
(g) all
(h) all
(i) all
(j) all
(k) all
(l) all
(m) all
(n) all
(o) all
(create a name!)
0
number(s) that are 2 units to the right of 0
number(s) that are 2 units to the left of 0
number(s) that are 2 units from 0
positive number(s) that are 2 units from 0
negative number(s) that are 2 units from 0
number(s) that are more than 2 units to the right of 0
number(s) that are more than 2 units to the left of 0
number(s) that are more than 2 units from 0
positive number(s) that are more than 2 units from 0
negative number(s) that are more than 2 units from 0
number(s) that are less than 2 units to the right of 0
number(s) that are less than 2 units to the left of 0
number(s) that are less than 2 units from 0 (include 0)
positive number(s) that are less than 2 units from 0
negative number(s) that are less than 2 units from 0
copyright Dr. Carol JVF Burns
331
EXERCISES
2.
Describe the numbers shaded below, using phrases similar to those in the
previous exercise. Assume that k is a positive number.
•
(a)
•
−3
(b)
..
....................
..
0
.....
..... .....
......
.....
..... .....
......
−3
(c)
0
......
..... ....
......
0
•
(d)
−k
..
....................
..
(f)
k
......
..... ....
......
0
......
..... ....
......
−k
3
•
0
......
..... ....
......
−k
the notation ‘ |x|’
represents the distance
between x and 0
........
.........
........
3
......
..... ....
......
−3
(e)
3
........
.........
........
k
......
..... ....
......
0
k
The notation ‘ |x| ’ (the number x inside vertical bars) is used to represent the
distance between the number x and 0 . (You’ll be told how to read the symbol
‘ |x| ’ in just a moment.)
When you see the vertical bars ‘ |stuff | ’, your first thought should be distance
from 0.
The symbol ‘ |3| ’ represents the distance between 3 and 0 . Thus, |3| = 3 , since
the number 3 is 3 units from zero.
The symbol ‘ |−3| ’ represents the distance between −3 and 0 . Thus, |−3| = 3 ,
since the number −3 is 3 units from zero.
Notice that |3| and |−3| are both mathematical expressions—they are numbers.
EXERCISES
3.
Fill in the blanks:
(a) The symbol ‘ |4| ’ represents the distance between
Thus, |4| =
, since the number
is
(b) The symbol ‘ |−4| ’ represents the distance between
Thus, |−4| =
, since the number
is
and
.
units from zero.
and
.
units from zero.
The expression ‘ |x| ’ is called the absolute value of x. There are two common
definitions of |x| : a geometric definition (which follows), and an algebraic definition. The algebraic definition is presented in a future section.
DEFINITION
the absolute value of x ;
geometric definition
Let x be a real number. Then,
|x| = the distance between x and 0 .
The symbol |x| is read as the absolute value of x.
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332
reading
absolute values
aloud
EXERCISES
The symbol ‘ |3| ’ is read as ‘the absolute value of 3 ’.
The symbol ‘ | − 3| ’ is read as ‘the absolute value of −3 ’.
The symbol ‘ |2x − 1| ’ is read as ‘the absolute value of two ex minus one’.
4.
State how you would read the symbols aloud.
Then, classify as an expression or a sentence.
Give the truth value of any sentence: always true (T), always false (F), or
sometimes true/sometimes false (ST/SF).
If a sentence is ST/SF, then give a value of x for which is is true, and a
value of x for which it is false.
(a) | − 7|
(b) |1 − 3x|
(c) | − 3| > 2
(d) | − 3| < 2
(e) | − 3| = 3
(f) |3| = −3
(g) | − 3| = −3
(h) |x| = 3
(i) |x| = −3
opposites have
the same
distance from zero
Opposites have the same distance from zero, so opposites have the same absolute
value:
|3| = | − 3| ;
|4.2| = | − 4.2| ;
in general, |x| = | − x| for all real numbers x .
distance is always
greater than
or equal to zero
Whenever you report distance, you give a number that is greater than or equal
to zero. You might say something like ‘this number is 2 units from 0’ , but you
would never say ‘this number is −2 units from 0’ .
Since |x| reports a distance, it follows that |x| is always greater than or equal
to zero. Thus, |x| ≥ 0 for all real numbers x . Recall that ‘nonnegative’ means
‘greater than or equal to zero’. Thus, |x| is a nonnegative quantity.
No matter what is inside the absolute value symbols, the resulting number is
always nonnegative. For example, | − x2 + 3x − 1| ≥ 0 for all real numbers x .
The observations in the preceding paragraphs are summarized as two important
properties of absolute value:
Properties of
Absolute Value
For all real numbers x ,
|x| ≥ 0 .
That is, absolute value is a nonnegative quantity.
For all real numbers x ,
|x| = | − x| .
That is, opposites have the same absolute value.
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333
Is a minus sign
inside
or
outside?
When working with absolute value expressions, you must pay attention to
whether a minus sign is inside the absolute value or outside, because it makes
a big difference!
A minus sign outside an absolute value denotes multiplication by −1 . For
example, −|3| = (−1) · |3| = (−1) · 3 = −3 .
The position of the minus sign affects the order of operations:
•
the expression ‘ | − 3| ’ denotes the sequence: start with the number 3 , first
take its opposite, and then find the resulting number’s distance from zero.
• the expression ‘ −|3| ’ denotes the sequence: start with the number 3 , first
find its distance from zero, and then multiply this result by −1 .
Notice that 3 , |3| , and | − 3| are all positive numbers.
However, −3 , −|3| , and −| − 3| are all negative numbers.
EXERCISES
5.
For these exercises, suppose that x is a nonzero number. That is, x might
be positive and x might be negative, but x is not 0 .
Decide if each of the following is always positive (P), always negative (N),
or sometimes positive/sometimes negative (SP/SN).
If an expression is SP/SN, then give a value of x for which it is positive,
and a value of x for which it is negative.
(a) |5|
(b) | − 5|
(c) −|5|
(d) −| − 5|
(e) x
(f) |x|
(g) −x
(h) | − x|
(i) 2x
(j) |2x|
(k) −2x
(l) | − 2x|
(m) −|2x|
(n) −| − 2x|
Next, the mathematical word ‘or’ is reviewed, and then some new notation is
introduced. This is preparation for solving sentences involving absolute value,
which is the subject of the next section.
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copyright Dr. Carol JVF Burns
334
review:
the mathematical
word ‘or’
Recall that the ‘or’ sentence ‘A or B’ is true when at least one of the subsentences is true: when A is true, or B is true, or both A and B are true.
Thus, ‘ x = 3 or x = −3 ’ is true for the values of x shaded below:
•
−3
•
0
3
You need to shade the value that makes the subsentence ‘ x = 3 ’ true, together
with the value that makes the subsentence ‘ x = −3 ’ true.
As a second example, ‘ x > 3 or x < −3 ’ is true for the values of x shaded
below:
..
....................
..
......
..... ....
......
−3
......
..... ....
......
0
........
.........
........
3
You need to shade the values that makes the subsentence ‘ x > 3 ’ true, together
with the values that makes the subsentence ‘ x < −3 ’ true.
EXERCISES
6.
Shade the value(s) of x that make each sentence true:
(a) x = 2
(b) x = −2
(c) x = 2 or x = −2
(d) x > 2
(e) x < −2
(f) x > 2 or x < −2
(g) x = 2 or x ≤ −2
(h) x = −2 or x ≥ 2
(i) x − 1 = 3 or x − 1 = −3
the sentence
‘ x = ±2 ’
is shorthand for
‘ x = 2 or x = −2 ’
The sentence: ‘ x = ±2 ’
is read as: ‘x equals plus or minus 2 ’
and is a shorthand for: ‘ x = 2 or x = −2 ’ .
That is, the symbol ‘ ± ’ is short for ‘plus or minus,’ using the mathematical
word ‘or’.
Thus, the sentence ‘ x = ±2 ’ is true for the values of x shaded below:
•
−2
EXERCISES
7.
•
0
2
Shade the value(s) of x that make each sentence true:
(a) x = ±3
(b) x = ±3 or x > 4
(c) x = 0 or x = ±3
(d) x = ±3 or x = ±4
solving sentences
like 2x − 1 = ±5 ;
two possible approaches
(create a name!)
When working with sentences involving ‘ ± ’ , you have two choices: break into
an ‘or’ sentence immediately, or wait until the last step to break into an ‘or’
sentence. Here’s an example of each approach:
copyright Dr. Carol JVF Burns
335
first approach:
SOLVE: 2x − 1 = ±5
break into an ‘or’
sentence immediately
FIRST APPROACH: Break into an ‘or’ sentence immediately:
2x − 1 = ±5
2x − 1 = 5 or 2x − 1 = −5
2x = 6 or 2x = −4
x = 3 or x = −2
second approach:
SOLVE: 2x − 1 = ±5
wait until the last step
to break into
an ‘or’ sentence
SECOND APPROACH: Wait until the last step to break into an ‘or’ sentence:
2x − 1 = ±5
2x = ±5 + 1
±5 + 1
x=
2
5+1
−5 + 1
x=
or x =
2
2
x = 3 or x = −2
you should understand
both approaches;
then, use whichever
is most comfortable
EXERCISES
You should understand both approaches. Then, use whichever approach is
most comfortable for you. Notice that, in both approaches, you are writing a
nice clean list of equivalent sentences. Also notice how you are using both the
Addition and Multiplication properties of equality to replace an equation with
an equivalent equation.
8.
Solve each sentence twice, using each of the two approaches just discussed.
Which way is most comfortable for you?
(a) 3x + 1 = ±2
(b) 2 − 5x = ±1
next section:
solving sentences
involving
absolute value
The next section presents tools for solving absolute value sentences such as
these:
|x| = 5
|x| < 5
|x| > 5
|x| ≥ 5
|2x − 1| ≥ 5
|3x − 5| < 4
5 − |1 + 3x| = 7
You’ll see that the process is simple, providing you keep in mind the critical
fact:
FF
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ABSOLUTE VALUE GIVES DISTANCE FROM 0
copyright Dr. Carol JVF Burns
FF
336
EXERCISES
web practice
Go to my homepage http://onemathematicalcat.org and navigate to my
Algebra I course, which has about 170 sequenced lessons. It can be used as
a complete year-long high school course, or one semester in college. You’re
currently looking at the pdf version—you’ll see that the HTML version has
unlimited, randomly-generated, online and offline practice in every section. It’s
all totally free. Enjoy!
SOLUTIONS TO EXERCISES:
INTRODUCTION TO ABSOLUTE VALUE
•
1. (a) all number(s) that are 2 units to the right of 0 :
−2
(b) all number(s) that are 2 units to the left of 0 :
0
2
0
2
•
−2
•
(c) all number(s) that are 2 units from 0 :
•
−2
0
−2
0
2
0
2
2
•
(d) all positive number(s) that are 2 units from 0 :
(e) all negative number(s) that are 2 units from 0 :
•
−2
(f) all number(s) that are more than 2 units to the right of 0 :
......
..... ....
......
−2
(g) all number(s) that are more than 2 units to the left of 0 :
...
...................
..
..
.....................
..
2
0
2
.....
..... .....
.....
−2
(h) all number(s) that are more than 2 units from 0 :
0
...
...... .....
.......
−2
...
...... .....
.......
0
(i) all positive number(s) that are more than 2 units from 0 :
−2
(j) all negative number(s) that are more than 2 units from 0 :
0
2
0
2
...
...... .....
.......
...
...... .....
.......
−2
0
2
.....
..... ....
......
.....
..... ....
......
−2
0
(k) all number(s) that are less than 2 units to the right of 0 :
(m) all number(s) that are less than 2 units from 0 (include 0) :
.....
..... .....
......
−2
(create a name!)
copyright Dr. Carol JVF Burns
2
.....
..... .....
......
0
2
.......
..... .....
....
.......
..... .....
....
−2
0
2
.....
..... ....
......
.....
..... ....
......
−2
0
(n) all positive number(s) that are less than 2 units from 0 :
(o) all negative number(s) that are less than 2 units from 0 :
........
.........
........
.....
..... .....
......
−2
(l) all number(s) that are less than 2 units to the left of 0 :
........
.........
........
2
......
..... ....
......
..
....................
..
........
.........
........
2
337
2. (a) all numbers that are 3 units from 0
(b) all numbers that are more than 3 units from 0
(c) all numbers that are less than 3 units from 0
(d) all numbers that are k units from 0
(e) all numbers that are more than k units from 0
(f) all numbers that are less than k units from 0
3. (a) The symbol |4| represents the distance between 4 and 0 . Thus, |4| = 4 , since the number 4 is 4 units
from zero.
(b) The symbol | − 4| represents the distance between −4 and 0 . Thus, | − 4| = 4 , since the number −4 is
4 units from zero.
4. (a) | − 7| : the absolute value of 7 ; expression (number)
(b) |1 − 3x| : the absolute value of one minus three ex; expression (number)
(c) | − 3| > 2 : the absolute value of negative three is greater than two; sentence; T
(d) | − 3| < 2 : the absolute value of negative three is less than two; sentence; F
(e) | − 3| = 3 : the absolute value of three is equal to three; sentence; T
(f) |3| = −3 : the absolute value of three is equal to negative three; sentence; F
(g) | − 3| = −3 : the absolute value of negative three is equal to negative three; sentence; F
(h) |x| = 3 : the absolute value of ex is equal to three; sentence; ST/SF; true for x = 3 or x = −3 , false
otherwise
(i) |x| = −3 : the absolute value of ex is equal to negative three; sentence; always false
5. (a) |5| is positive (|5| = 5)
(b) | − 5| is positive (| − 5| = 5)
(c) −|5| is negative (−|5| = −5)
(d) −| − 5| is negative (−| − 5| = −5)
(e) x is SP/SN; positive for x = 3 , negative for x = −3
(f) |x| is always positive
(g) −x is SP/SN; positive for x = −3 (−x = −(−3) = 3) and negative for x = 3 (−x = −3)
(h) | − x| is always positive
(i) 2x is SP/SN; positive for x = 1 and negative for x = −1
(j) |2x| is always positive
(k) −2x is SP/SN; positive for x = −1 (−2x = −2(−1) = 2) and negative for x = 1 (−2x = −2(1) = −2)
(l) | − 2x| is always positive
(m) −|2x| is always negative
(n) −| − 2x| is always negative
•
6. (a) x = 2
−2
(b) x = −2
2
0
2
•
−2
(c) x = 2 or x = −2
0
•
−2
•
0
(d) x > 2
−2
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2
........
..... ....
........
copyright Dr. Carol JVF Burns
0
........
.........
........
2
338
(e) x < −2
...
...................
...
(f) x > 2 or x ≤ −2
..
...................
...
(g) x = 2 or x ≤ −2
..
...................
...
........
..... ....
........
−2
0
2
•
........
..... ....
........
−2
0
2
•
•
−2
0
2
•
(h) x = −2 or x ≥ 2
........
.........
........
•
−2
0
........
.........
........
2
(i) x − 1 = 3 or x − 1 = −3 : rewrite as the equivalent sentence
•
x = 4 or x = −2 :
•
−2
0
•
7. (a) x = ±3
•
−3
0
3
•
(b) x = ±3 or x > 4
−3
0
•
(c) x = 0 or x = ±3
−3 −2 −1
(d) x = ±3 or x = ±4
4
• •
−4 −3
•
.....
..... .....
.....
3
4
•
0
........
.........
........
•
1
2
3
• •
0
3
4
8. (a)
3x + 1 = ±2
3x + 1 = 2 or 3x + 1 = −2
3x = 1 or 3x = −3
1
or x = −1
x=
3
3x + 1 = ±2
3x = ±2 − 1
±2 − 1
x=
3
2−1
−2 − 1
x=
or x =
3
3
1
x=
or x = −1
3
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copyright Dr. Carol JVF Burns
339
(b)
2 − 5x = ±1
−5x + 2 = ±1
−5x + 2 = 1 or − 5x + 2 = −1
−5x = −1 or − 5x = −3
1
3
x=
or x =
5
5
2 − 5x = ±1
−5x + 2 = ±1
−5x = ±1 − 2
±1 − 2
x=
−5
1−2
−1 − 2
x=
or x =
−5
−5
1
3
x=
or x =
5
5
(create a name!)
copyright Dr. Carol JVF Burns
340